Question: Valeria is playing with her accordion. The length of the accordion $A(t)$ (in $\text{cm}$ ) after she starts playing as a function of time $t$ (in seconds) can be modeled by a sinusoidal expression of the form $a\cdot\cos(b\cdot t)+d$. At $t=0$, when she starts playing, the accordion is $15\text{ cm}$ long, which is the shortest it gets. $1.5$ seconds later the accordion is at its average length of $21\text{ cm}$. Find $A(t)$. $\textit{t}$ should be in radians. $A(t) = $
Solution: The strategy First, we should convert the given information about the real-world context into mathematical terms of the sinusoidal function and its graph. Then, we should use the given information to find the amplitude, midline, and period of the function's graph. Finally, we should find $a$, $b$, and $d$ in the expression $a\cos(b\cdot t)+d$ by considering the features we found. Converting the given information into mathematical terms At $t=0$, the accordion is $15\text{ cm}$ long. This means the graph of the function passes through $(0,15)$. We are given that this is the accordion's shortest length, which corresponds to a minimum point of the graph. $1.5$ seconds later (which means $t=1.5$ ) its length is $21\text{ cm}$. This corresponds to the point $(1.5,21)$. We are given that this is the average length, which corresponds to the midline of the graph. In conclusion, the graph has a minimum point at $(0,15)$ and then intersects its midline at $(1.5,21)$. Determining the amplitude, midline, and period The midline intersection is at $y={21}$, so this is the midline. The minimum point is $6$ units below the midline, so the amplitude is ${6}$. The minimum point is $1.5$ units to the left of the midline intersection, so the period is $4\cdot 1.5={6}$. [Why did we multiply by 4?] Determining the parameters in $a\cos(b\cdot t)+d$ Since the minimum at $t=0$ is followed by a midline intersection, we know that $a<0$. [How do we know that?] The amplitude is ${6}$, so $|a|={6}$. Since $a<0$, we can conclude that $a=-6$. The midline is $y={21}$, so $d=21$. The period is ${6}$, so $b=\dfrac{2\pi}{{6}}=\dfrac{\pi}{3}$. The answer $A(t)=-6\cos\left(\dfrac{\pi}{3}t\right)+21$